Several methods are used to evaluate the functions depending on the arguments aa, bb and xx. The methods include Wise's asymptotic expansion (see Wise (1950)) when a > ba>b, continued fraction derived by DiDonato and Morris (1992) when aa, b > 1b>1, and power series when b ≤ 1b≤1 or b × x ≤ 0.7b×x≤0.7. When both aa and bb are large, specifically aa, b ≥ 15b≥15, the DiDonato and Morris (1992) asymptotic expansion is employed for greater efficiency.

Once either Ix(a,b)Ix(a,b) or Iy(b,a)Iy(b,a) is computed, the other is obtained by subtraction from 11. In order to avoid loss of relative precision in this subtraction, the smaller of Ix(a,b)Ix(a,b) and Iy(b,a)Iy(b,a) is computed first.

Accuracy

nag_specfun_beta_incomplete (s14cc) is designed to maintain relative accuracy for all arguments. For very tiny results (of the order of machine precision or less) some relative accuracy may be lost – loss of three or four decimal places has been observed in experiments. For other arguments full relative accuracy may be expected.